A simple heuristic proof of Hardy and Littlewood's conjecture B

"Proof" We begin by considering the following combinatorial problem: Let A = {1, 2,3, .. ., a} and choose from A a subset B consisting of b elements. Now, say we choose another subset B' from A consisting of b' elements. The elements in B' are chosen randomly (by randomly I mean that every element of A has an equal chance of being picked). Then, the expected number of elements in B n B' is equal to b' * (b/a) = b'b/a. Since this trivial result is used throughout this article, we refer to it as (1). We also reference Dirchlet's Theorem (see [1] for a proof of Dirichlet's Theorem) which states that given an arithmetical progression ak + b, where (a, b) = 1, then